Physics for Scientists and Engineers: A Strategic Approach with Modern Physics (3rd Edition)

Published by Pearson
ISBN 10: 0321740904
ISBN 13: 978-0-32174-090-8

Chapter 38 - Quantization - Exercises and Problems - Page 1153: 34

Answer

No.

Work Step by Step

We know that the prediction of the wavelengths in the hydrogen absorption spectrum is given by $$\lambda_{m\rightarrow n}=\dfrac{\lambda_0}{\dfrac{1}{m^2}-\dfrac{1}{n^2}}$$ where $m=1,2,3,..$, and $n=m+1,m+2,m+3,...$ We know that most of the atoms in a gas are in the ground state which means that the only quantum jumps seen in the absorption spectrum start from this state. Thus, let's set $m=1$, and hence $n=2,3,4,...$ $$\lambda_{1\rightarrow n}=\dfrac{\lambda_0}{\dfrac{1}{1^2}-\dfrac{1}{n^2}}$$ where $\lambda_0=91.18$ nm $$\lambda_{1\rightarrow n}=\dfrac{91.18}{1-\dfrac{1}{n^2}}$$ Now let's try to find the absorption spectral line with 656.5 nm. $$\lambda_{1\rightarrow 2}=\dfrac{91.18}{1-\dfrac{1}{2^2}}=\color{green}{\bf 121.6}\;\rm nm$$ $$\lambda_{1\rightarrow 3}=\dfrac{91.18}{1-\dfrac{1}{3^2}}=\color{green}{\bf 102.5}\;\rm nm$$ $$\lambda_{1\rightarrow 4}=\dfrac{91.18}{1-\dfrac{1}{3^2}}=\color{green}{\bf 97.3}\;\rm nm$$ We can see that the wavelengths decrease by increasing $n$ which means that we do not find any spectral line with $\lambda=656.5$ nm. Therefore, no spectral line with wavelength 656.5 nm is seen in the absorption spectrum of hydrogen atoms.
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